Eigenvalues of the bi-Xin-Laplacian on complete Riemannian manifolds

Xiaotian Hao; Lingzhong Zeng; Xiaotian Hao; Lingzhong Zeng

doi:10.3934/cam.2023009

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 162-176. doi: 10.3934/cam.2023009

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Research article

Eigenvalues of the bi-Xin-Laplacian on complete Riemannian manifolds

Xiaotian Hao ¹,
Lingzhong Zeng ^{1,2
,
,}

1.
School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China
2.
Jiangxi Provincial Center for Applied Mathematics, Jiangxi Normal University, Nanchang 330022, China

Received: 06 December 2022 Revised: 17 March 2023 Accepted: 26 March 2023 Published: 05 May 2023
35P15, 53C40

The clamped plate problem describes the vibration of a clamped plate in the classical elastic mechanics, and the Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short). In this article, we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space. On one hand, we obtain some eigenvalue inequalities of the bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions. Let us emphasize that, this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds, some types of warp product manifolds and homogenous spaces. On the other hand, we also consider the eigenvalue problem of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality. In particular, we can give an estimate for the lower order eigenvalues on the cylinders.
- bi-Xin-Laplacian,
- eigenvalue,
- Riemannian manifold,
- Cartan-Hadamard manifold,
- homogenous manifold,
- cylinder
Citation: Xiaotian Hao, Lingzhong Zeng. Eigenvalues of the bi-Xin-Laplacian on complete Riemannian manifolds[J]. Communications in Analysis and Mechanics, 2023, 15(2): 162-176. doi: 10.3934/cam.2023009

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Abstract

The clamped plate problem describes the vibration of a clamped plate in the classical elastic mechanics, and the Xin-Laplacian is an important elliptic operator for understanding the geometric structure of translators of mean curvature flow(MCF for short). In this article, we investigate the clamped plate problem of the bi-Xin-Laplacian on Riemannian manifolds isometrically immersed in the Euclidean space. On one hand, we obtain some eigenvalue inequalities of the bi-Xin-Laplacian on some important Riemannian manifolds admitting some special functions. Let us emphasize that, this class of manifolds contains some interesting examples: Cartan-Hadamard manifolds, some types of warp product manifolds and homogenous spaces. On the other hand, we also consider the eigenvalue problem of the bi-Xin-Laplacian on the cylinders and obtain an eigenvalue inequality. In particular, we can give an estimate for the lower order eigenvalues on the cylinders.

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